Takashi Kumagai studied at Kyoto University, where he defended his PhD thesis in 1994 (supervisor: Shinzo Watanabe). After working at Osaka University and Nagoya University, he went back to Kyoto University in 1998. He is now a professor at the Research Institute for Mathematical Sciences (RIMS), Kyoto University. His research areas are anomalous diffusions on disordered media such as fractals and random media, and potential theory for jump processes on metric measure spaces. He gave St. Flour 2010 lectures, was a plenary speaker at SPA 2010 in Osaka, invited speaker at the International Congress of Mathematicians in Seoul 2014.

There has been a long history of research on heat kernel estimates and Harnack inequalities for diffusion processes. Harnack inequalities and Hölder regularities for harmonic/caloric functions are important components of the celebrated De Giorgi-Nash-Moser theory in harmonic analysis and partial differential equations. In early 90’s, equivalent characterizations for parabolic Harnack inequalities (that is, Harnack inequalities for caloric functions) that are stable under perturbations were obtained by Grigor’yan and Saloff-Coste independently, and later extended in various directions.

Such stability theory has been developed only recently for symmetric jump processes, despite of the fundamental importance in analysis. In 2002, Bass and Levin obtained heat kernel estimates for Markov chains with long-range jumps on the d-dimensional lattice. Motivated by the work, Chen and Kumagai (2003) obtained two-sided heat kernel estimates and parabolic Harnack inequalities for symmetric stable-like processes, which are perturbations of symmetric stable processes, on Ahlfors regular subsets of Euclidean spaces. The results include equivalent stable condition for the two-sided stable type heat kernel estimates. There has been vast amount of work related to potential theory on symmetric stable-like processes since around that time. Definite answers are given in the recent trilogy by Chen, Kumagai and Wang on the stability of heat kernel estimates and parabolic Harnack inequalities for symmetric jump processes on general metric measure spaces. While both of them are stable under perturbations, unlike diffusion cases, heat kernel estimates are not equivalent to parabolic Harnack inequalities for jump cases.

In the talk, I will summarize developments of the De Giorgi-Nash-Moser theory for symmetric jump processes and discuss its applications. The applications include discrete approximations of jump processes and random media with long-range jumps. The talk is based on joint works with my collaborators; M.T. Barlow, R.F. Bass, Z.-Q. Chen, A. Grigor’yan, J. Hu, P. Kim, M. Kassmann and J. Wang.

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